To summarize, that is:
1.To get a universal view of Inference and Regression and if time allow, also try to get a universal view of the mathematics that I’ve learned.
2.To gain deeper understanding at some particular point, if do not have enough time, at least get some idea what view that way would lead to(Usually, here you might also need to stand at some other places to look at it). It’s not possible to fulfill this without seeing some egs and work out some problems by myself.
3.Form good habits of thinking and learning. For instance, the appetite of solving problems and the motivation to investigate.
4.Avoid all unnecessary events, submerge in your own world and live with strength!
5.Now I have a feeling that if you can not maintain consistency and intensity at the same time, then consistency is preferable to intensity under most occasions. So, be consistent, either in study and in love, i.e. if you determine to learn from one book, then you should follow that book(or at least that related part in that book) thoroughly and resist the temptation to pick up another one in the middle.
The math that I felt more and more important to my current research field:
1. Analysis, i.e. Integration techniques, Taylor expansion(in high dim, Hessian matrix~), Conditional Optimality, some series and function convergence examples, Modes of convergence
And I found the need to learn complex analysis, like its integration theory and analytic theory(one eg is the integration of (sinx/x)^2). The other important discipline is Fourier analysis, I need to get a sufficient understanding in this field. Like the space structure, Fourier transform, FFT, the idea of Kernel and smoothing etc. If I do this, I should be able to tell why Fourier analysis plays an important role in probability, or in particular, time series study.
2. Algebra, i.e. Linear transf and its geometric meaning(like real symm, Householder transf, Givens. Here you definitely need to understand some spectral theory of matrix), unitary space and the idea of projection and orthogonality(note the conditional expectation!), matrix decomposition(QR, SVD, Triangular, Polar, etc), matrix diagonalization(also consider the partition matrices), matrix canonical forms, Quadratic forms and its relationship with ellipse and other geometry objects( This also enable me to see the corresponding application in stat), etc.
For abstract algebra, its notion and idea also plays important role in some branches of stat, say, invariant theory and of course, the matrix and space theory behind regression and time series.
And some stat methods and algorithms I need to know:
2. Gradient descent( and the stochastic version)
3. PCA, LDA
4. Idea of classification
6. LASSO, LARS
7. Comparison about Frequentist and Bayesian approach in stat(so, read that Springer book!)